Talk:Spline (mathematics)

Bezier curve not a spline ?

I did a complete rewrite of the article. A Bezier curve is not a spline. MathMartin 18:02, 19 Sep 2004 (UTC)

What is the difference? 20 Mar 05

To explain what a spline is I think it is best to contrast splines with polynomials. Splines are then defined as piecewiese polynomials (of course you can consider a polynomial a spline with only one piece). After this difference is clear you can discuss different forms of the polynomials used to construct the spline (like Bernstein form, Hermite form, Monomial form etc.).

So although in some sense a Bezier curve is a spline I think it is clearer not use it as a central example in the main spline article. MathMartin 16:04, 21 Mar 2005 (UTC)

I strongly agree. A Bezier curve may be a smaller portion of a spline, and one might argue it is a spline made of one curve, but it is a horrible example for the article 'Spline'. Mgsloan 23:13, 15 March 2007 (UTC)

A Bezier curve is simply an image of a parametric polynomial. One may argue on strictly technical grounds that it is a piece-wise polynomial spline consisting of one piece, but I find such arguments more clever than informative to the reader. There is a better image than Image:Splined epitrochoid.png which does illustrate a piece-wise degree three polynomial spline and a piece-wise degree-one curve, each approximating a single degree-six curve. While it was drawn for a different purpose, I find it less misleading than the present image, and will put it in place

A Bezier curve is, first and foremost, a curve rather than a (scalar-valued) function; it is, more precisely, a way to represent certain curves in way that makes retrieval of certain information easy and also helps when putting together smooth curves from parametrit polynomial pieces. I agree that any curve as an illustration in an article on spline functions is misleading, as the distinction between curves and (scalar-valued) functions is a fundamental one. Deboor (talk) 03:51, 26 June 2011 (UTC) after I post this note. Gosgood (talk) 02:08, 26 June 2008 (UTC)

History section

The greater part of the section History is copied verbatim from a post to the NA Digest. Can anybody confirm that we have permission to do this? I also asked the anonymous contributor for clarification at User talk:205.250.40.97. -- Jitse Niesen (talk) 16:16, 6 October 2005 (UTC)

Resolved after some e-mails. -- Jitse Niesen (talk) 12:51, 18 October 2005 (UTC)

Definition: Closed subintervals vs. Half-open subintervals

In the Definition, some people want to use closed subintervals $[t_{i},t_{i+1}]$ of $[a,b]$ (option 1), while others want to use half-open subintervals $[t_{i},t_{i+1})$ of $[a,b]$ (option 2a) or of $[a,b)$ (option 2b).

The differences between these options are NOT merely cosmetic! It is important to note why (1) is so mathematically different from (2a) and (2b). Under (1), any two consecutive subintervals will share a knot, so they are not disjoint. In other words, all subintervals (together) do not constitute a partition of $[a,b]$, but only a covering (and not a packing). All this implies that two neighboring polynomial pieces will, under (1), always match continuously over the knot in common. This will exclude step functions, for example, or any spline with discontinuities for that matter.

If we are to allow discontinuous splines, we are motivated to use half-open subintervals. Option (2b) would then be the most rigorous, while (2a) offers an attractive compromise between (1) and (2b). 213.103.121.83 (talk) 15:05, 10 November 2006 (UTC)

Abstract: Subject Classification

The abstract used to start with "In the mathematical subfield of numerical analysis, ...", and "In the computer science subfields of computer-aided design and computer graphics, ...".

Apparently this formulation was not clear to all, since an editor [07:55, 18 October 2006 211.29.178.155] changed "subfield" to "field" since it "Doesn't make sense to talk of a *sub*field without mentioning a larger encompassing field!".

Since I contributed the original formulation, let me try to clarify what I intended by it. I thought it was clear that the "encompassing fields" were "mathematics" on the one hand and "computer science" on the other, while the SUBfields were "numerical analysis" on the one hand, and "computer-aided design" and "computer graphics" on the other. If anyone knows how to formulate this more clearly, please feel free to post your proposition.

For your information, these (sub)fields were taken from the 2000 Mathematical Subject Classification (MSC2000) of the American Mathematical Society (AMS): See http://www.ams.org/msc/

Definition: Knots are they points, values, vertices, vectors, or ... ?

Since this question was raised by an editor [09:41, 28 September 2006 SpaceDude (knots are not points, they are scalar values? would like confirmation on this... how can inequality be used on points?)], let me try to provide a short answer.

In the article's definition of splines, knots are elements of $\mathbb {R}$. Whether you call them "points", "vectors", "values", or ... depends on what you mean by $\mathbb {R}$ in the first place.

If you equip the set $\mathbb {R}$ with the usual structure of an affine space, you can call the knots "points", to emphasize the geometric view of $\mathbb {R}$ as a manifold. You can then consider "weighted averages" of knots. Such averages are needed by the "de Boor algorithm".

If you equip the set $\mathbb {R}$ with the usual real vector space structure, you can call the knots "vectors", to emphasize the differential-geometric view of $\mathbb {R}$ as a tangent space/line. You can then consider "sums" and "real multiples" of knots. Very useful for uniform partitions and Fourier techniques to construct B-splines ("box splines").

If you equip the set $\mathbb {R}$ with the usual structure of a field, you can call the knots "values", to emphasize the algebraic view of $\mathbb {R}$ as a space of numbers. You can then consider "sums", "differences", "products" and "quotients" of knots. Very useful for the divided difference approach to B-splines.

If you equip the set $\mathbb {R}$ with the usual ordering structure, you could call the knots "vertices" or "exposed points", to empasize the convexity of the subintervals. You can then consider "in-betweenness" of knots and whether any knot is larger than another one. Needed by polyhedron projection techniques to construct B-splines geometrically ("polyhedral splines", "box splines", "simplex splines", "cone splines").

And so on ...

Now, the set $\mathbb {R}$ is frequently assumed to be equiped with SEVERAL such structures at the same time -- this is not contradictory as long as the structures are compatibile with each other (which they are), so they can be combined meaningfully. This is how "inequalities" can be used on "points": take $\mathbb {R}$ with the usual affine structure and the usual ordering structure combined.

But the assumed (combined) structure(s) are not always stated explicitly though. Fortunately, it is not difficult to work backwards and recover the needed structure(s) on $\mathbb {R}$ from the way the elements of $\mathbb {R}$ are used.

So what are they then, these knots. Are they points? vectors? values? vertices? ... It depends on the writer.

My impression is that computer scientists prefer to maximally equip $\mathbb {R}$ with all structures possible, so they can do what they want with its elements (without having to bother about what they mean by $\mathbb {R}$) and can call them how they want: "vectors", "values", "points", ... (Similarly, in "3D" they can speak of "points" and "vectors" -- although CAGD writers definitely prefer "vectors").

Mathematicians, out of economy and clarity, would equip $\mathbb {R}$ only minimally: they wouldn't give it a particular structure unless it is really needed.

For example, in CAGD text books one speaks of "vectors" even if the origin has no special role, so (affine) "points" would do for mathematicians.

To conclude, knots in $\mathbb {R}$ can be called "vectors", "values", "points", "vertices", ...

Nevertheless, the term "value" has the disadvantage that it is (so far) not used in multivariate spline theory (not discussed here).

To my way of thinking, knots are parameters connected with the representation of splines as a weighted sum of B-splines. Since B-splines are not mentioned in this article, knots should also not be mentioned. The points where two polynomial pieces join used to be called (e.g., by Birkhoff) "joints" but, these days, the term 'breakpoint' or 'break' is used for such a locus. Use of 'points' for them should be discouraged since one usually deals with data points and only their first coordinate, the data site, gives rise to a break in cubic spline interpolation, while its second coordinate, the data value, is the value one hopes to match at that site by the interpolating spline. Deboor (talk) 03:37, 26 June 2011 (UTC)

Yes, there are many structures possible on $ \mathbb{R} $ and I am sure that there are applications that require each. However, it is not clear in the article what underlying set a "knot" belongs to, nor are they defined anywhere. This should be fixed. 128.32.92.238 (talk) 19:54, 23 September 2013 (UTC)

examples

Hi the example is ok but need to explicitly show the smoothness vector $\mathbf {r}$. SNx 20:11, 4 December 2006 (UTC)

Spline smoothness

In the example section, it is said that second derivative should be set to 0 at end points so that the spline be a straight line outside of the interval, while not disrupting its smoothness. This forces the spline to be straight at end points, but to force smoothness, one need to get the same slope on both sides of the end points, so $S'_{1}(a)\,=S'_{2}(a)$ where $S_{1}$ and $S_{2}$ stands for the two polynomials each side of $a$. 132.246.2.25 21:32, 23 July 2007 (UTC)Matthieu Deveau

It is mentioned that the natural cubic spline is of "continuity" C². However, previously in the text, the term "smoothness" has always been used for the measure Cⁿ. If these two are synonymous, the slope equality is thus already assumed. It wouldn't hurt, however, mentioning it explicitly.

So: are smoothness and continuity equal? If so, should both terms be used where Cⁿ is defined? If not, I've misunderstood and that is not a good sign ;) Beryllium-9 (talk) 20:35, 19 August 2008 (UTC)

Polynomials are in C^$\infty$

Expressing the connectivity as a "loss of smoothness" is reasonable, since if S were a simple polynomial throughout a neighborhood of t_i, it would have smoothness Cⁿ at t_i, and you would expect to lose smoothness in order to break a polynomial apart into pieces.

Every polynomial is in $C^{\infty }$ so it would have smoothness $C^{\infty }$ at t_i.

However, it's a fact that for 2 polynomials of degree n it's enough their 0 to n derivatives are equal at point p, for these polynomials to be the same one poly. in fact (hence, to be $C^{\infty }$ smooth at point p). Eventually, I understand the idea behind n-"number of knots" loss of smoothness, but it was a bit misleading at first. Any ideas how to improve that? SalvNaut 21:03, 27 September 2007 (UTC)

Clarification

I'm not competent to edit the content but in Definition section C^r is introduced and while there is some clarification about what it means, there is nothing about what it IS (the set of all functions with derivatives of order 0 to r). Shouldn't definition section define terms the common reader might be unfamiliar with? Also, the comment that polynomials are in C^$\infty$ makes the statement that the smoothness is at most n-r, with r = $\infty$ really confusing to us poor laymen --what does a negative smoothness mean? Might be best to simply remove C and just talk about derivatives to order r. Introducing C doesn't seem to be necessary. Just a thought.71.31.154.4 (talk) 16:57, 1 November 2009 (UTC)

Definition Section Clarification

I added the clarification tag on the Definition section. A huge number of symbols and variables are introduced in the first few sentences. Even if these are technically rigorously defined (which I don't think they are), some English explanation of what the mathematical statements mean are necessary to improve readability, especially for someone who is being introduced to splines for the first time. Even the use of the blackboard bold R being used to denote "the set of all real numbers" is just a convention that might be clarified by including a simple entry like "where R is the set of all real numbers" in a list of variable definitions. mcs (talk) 16:18, 21 August 2008 (UTC)

Spline under Tension

I can't find any mention of the univariate spline under tension anywhere on Wikipedia, there is a concise presentation at the beginning of "A method for construction of surfaces under tension" by Nielson and Franke (sorry, no link). This is a very nice generalization of the "natural cubic spline" and piecewise linear interpolation, and the paper mentioned is referenced by "Terrain Simulation Using a Model of Stream Erosion" by Kelley, Malin, and Nielson. Am I not looking in the right place? 136.152.138.22 (talk) 20:12, 17 September 2008 (UTC)

Algorithm

Can we discuss validity and maybe clarifying the algorithm I posted? I am pretty sure it is correct, but I am not a mathematician, and would like some expert opinion on it.

Also can somebody write an algorithm for computing cubic splines in 3d space? —Preceding unsigned comment added by Lev a neiman (talk • contribs) 05:25, 10 January 2009 (UTC)

hi Lev a neiman. I reviewed the algorithm, since I needed to implement it today, and I've found some small errors. I made the corrections to the algorithm a few minutes ago. In the following paragraphs I will give you an explanation regarding the changes I made and hope that they are valid. but first I have to ask you a question: from where did you get the algorithm? I'm just curious and I would be grateful to hear from you. so, let's move to the abovementioned explanation:

• at point 1., the algorithm mentioned to "create a new array a of size n and for i = 0,...,n-1 set $a_{i}=y_{i}$". since the number of coordinates is n+1, the size of the array a should be n+1 too, due to the fact that the algorithms intends to set $a_{i}=y_{i}$. the for-definition changes too, because we now need to iterate from i = 0,...,n.

• at point 2. I moved the initialization of the array c down to point 5. and initialized it with a size of n + 1, since at point 8. it was mentioned to set $c_{n}=0$. if the array is initialized with a size of n and we wanted to access c[n], we would get an index-out-of-bound-error. the size of n + 1 is also needed, because at point 9. the algorithm accesses $c_{j+1}$ and index j (initialized with n - 1) would cause an index-out-of-bound-error, since in that case it would want to access c[n] in the first iteration.

• at point 4. I changed the size for the array α (alpha) to n. this is due to the fact that the for-definition defines the boundary from 0,...,n-1 and in the following lines no forward-access is made that would cause an index-out-of-bound-error (e.g. α[i+1]).

• finally, at point 9. and 10. I changed the number of splines S to n and the boundary of the for-definition to 0,...,n-1. The first change was made due to the fact that with n + 1 coordinates you will get n splines and the second change is simply a result of the first one.

I hope my feedback and my changes to the algorithm are okay. if you have some suggestions or corrections, I would be pleased to hear from you. Giuseppe A (talk) 16:47, 6 May 2009 (UTC)

I changed c[n]=0 for c[n]=1 in step 8 because else you get a ZeroDivision error in step 9.1

I needed to implement algorithm for creating clamped cubic spline and I encountered problems in your algorithm. The most obvious fact is that in algorithm there are never used values of first derivation in end points ( at that point I decided to look somewhere else for this algorithm ). But it seems to me, that there are more problems. Place, where I found almost same algorithm, is: http://www.michonline.com/ryan/csc/m510/splinepresent.html But it is quite hard to read it on that page, (for example array α in your algorithm is there without name - members of this array are just denoted as corresponding lower index,) however I haven't found any mistake in that algorithm and my program based on it work fine (so far). I suggest to correct to according to mentioned page.

Hope my feedback will be useful.  —Preceding unsigned comment added by 184.144.70.128 (talk) 22:55, 26 December 2010 (UTC) 

I was looking through the algorithm and I agree that it needs some clarification to avoid confusion. The simple fact of the matter is that it's very poorly written with inconsistent indexing. For most of the arrays, they are of length n with indices from 0 to n-1, but for the set alpha it is indexed starting at 1. This is a major issue for people trying to implement this who might be relying on the index numbers, as it will result in the incorrect output. Kasaiwind (talk) 20:37, 21 October 2018 (UTC)

I've removed it for being unreferenced and apparently WP:OR, but there were too many issues with it and I'd have likely killed it off anyway. Pseudocode is a much easier way to express an algorithm; I've never seen it done in this way at all, probably because it's just painful; I've been programming in assembly language(s) for various architectures for 30 years, studied pure mathematics in college, and was a compiler engineer working on LLVM before I retired so I've written painfully long mathematical proofs, read even longer ones, sat around reading the ISO C++ standard for hours, digging through instruction encodings in arch manuals to find bugs, etc... I had no problems getting what I needed from those kind of tech manuals on the first try. Unlike wikipedia, those weren't targeted at the general public, or even very many programmers, they thankfully never have to deal with them... if they do, they'll find that an algorithm for a single instruction (don't worry, I picked an obnoxious AVX512 with a bitmask, using the EVEX version which is defined as

CVTTPS2DQ—Convert with Truncation Packed Single-Precision Floating-Point Values to Packed Signed Doubleword Integer Values

Opcode / Instruction: EVEX.512.F3.0F.W0 5B /r VCVTTPS2DQ zmm1 {k1}{z}, zmm2/m512/m32bcst {sae} (note the parameters, since the pseudocode uses them)

Op/En: B - 64/32 bit mode support V/V

CPUID Feature Flag: AVX512F

Description: Convert sixteen packed single-precision floating-point values from zmm2/m512/m32bcst to sixteen packed signed doubleword values in zmm1 using truncation subject to writemask k1.

VCVTTPS2DQ (EVEX encoded versions) when src operand is a memory source

(KL, VL) = (4, 128), (8, 256), (16, 512)

FOR j ← 0 TO 15

i ← j * 32

IF k1[j] OR *no writemask*

THEN

IF (EVEX.b = 1)

THEN

DEST[i+31:i] ← Convert_Single_Precision_Floating_Point_To_Integer_Truncate(SRC[31:0])

ELSE

DEST[i+31:i] ← Convert_Single_Precision_Floating_Point_To_Integer_Truncate(SRC[i+31:i])

FI;

ELSE

IF *merging-masking*  ; merging-masking

THEN *DEST[i+31:i] remains unchanged*

ELSE

DEST[i+31:i] ← 0  ; zeroing-masking

FI

FI;

ENDFOR

DEST[MAXVL-1:VL] ← 0

They list all of the possible inputs in the instruction definition, define a couple of standard helper variables at the beginning, etc. The only thing that probably looks weird, especially if you came from somewhere horrible like python, is the subscripting in the array registers. This is a processor and offsets in vector registers / memory they're looping through so those just define the bit range being set or read from, which is also handy for the broadcast case (EVEX.b) where they just leave the "i" out to indicate that the memory source doesn't change. KL isn't used in this pseudocode and VL is always 512 and they could have folded the other lenghts of EVEX encodable legacy instructions in but it would have created more conditionals and this kept the definition standardized throughout the manual... which is important since the combined manual for Intel (and I have the 2017 version before they added the AMX cores) is nearly 5000 pages long and everything is at least that detailed.

Something for a regular algorithm doesn't need to be anywhere near as difficult to look at, you don't need to use ranges in array subscripts and shouldn't, for something like this complex conditionals barely exist, you can define structures to make things readable by not arbitrarily switching properties of a thing from being properties to being subscripted elements written like matrix subscripts would be written in math because those aren't important, and the properties are fixed to the index and not vice versa so standard notation there makes it less confusing to see what's being done, you don't need to show for loops for hyper-basic things like copies at all when you can just do a[] = b[], or a[0...n-1] = b[0...n-1], etc. There's no need to say that a new array should be created because it probably shouldn't, or if it should that part will be obvious in the language it's in. But that's all for future reference since you can't just write something like this and put it on wikipedia, things are supposed to be referenced here despite this article doing one of the worst jobs of it I've ever seen throughout. Just a barely usable ref list at the bottom with only a few of them badly parenthesis ref'd in the text doesn't help a single person anywhere, then it's loaded up with math that probably needs to be thrown out because nobody could be bothered to list where it came from and nobody is going to go through the effort of digging up all these weird old books to try to find it when the symbols make it unsearchable even if a PDF magically appears.

I don't know why this particular article has turned into such weirdness but it needs the kind of rework I can't give it. --A Shortfall Of Gravitas (talk) 01:51, 13 February 2025 (UTC)

Control point

Control point (mathematics) redirects to this article but the article does not use the term "control point"; please define or use the term. --Una Smith (talk) 04:03, 30 September 2008 (UTC)

Hard to read

Annoying that the algorithm uses both letter a, and the symbol alpha, and the fonts (at least where I am) display both in a very similar fashion in parts of the formulas. —Preceding unsigned comment added by 66.205.73.36 (talk) 02:59, 21 January 2009 (UTC)

Reticulating Splines

No information on how one would go about reticulating them? Why not? We need to add this in right away. —Preceding unsigned comment added by Pomegranete (talk • contribs) 10:59, 5 September 2009 (UTC)

It was just a joke on the part of the Simcity people, which you know by the tone. --A Shortfall Of Gravitas (talk) 02:14, 13 February 2025 (UTC)

Propose section for deletion

The section Spline (mathematics)#Theoretical problem in random space should be deleted. It was added by user:Yuanfangdelang who has added a lot of similarly dubious material to several statistics articles. Skbkekas (talk) 02:50, 26 January 2010 (UTC)

This section has now been deleted. Skbkekas (talk) 20:30, 29 January 2010 (UTC)

"General expression for a C² interpolating cubic spline" seems incorrect

The equation for $S_{i}(x)$ in this section is for the cubic polynomial solving the boundary value problem:

• $S_{i}(t_{i}^{})=f^{}(t_{i}^{})$,
• $S_{i}^{\prime \prime }(t_{i})=f^{\prime \prime }(t_{i})$,
• $S_{i}(t_{i-1}^{})=f^{}(t_{i-1}^{})$,
• $S_{i}^{\prime \prime }(t_{i-1})=f^{\prime \prime }(t_{i-1})$

Notice that there is nothing about the first derivative $S_{i}^{\prime }(x)$, and in fact when used on non-trivial data the first derivatives do not match at the knots. Thus the resulting cubic is actually only C⁰, no better -- at least in terms of continuity -- than a linear interpolation. This section also seems to differ from the previous sections on the same page in that $S_{i}(x)$ is for $[t_{i-1},\ t_{i}]$ where earlier $S_{i}(x)$ is for $[t_{i},\ t_{i+1}]$. It appears that more than a year ago this had the closed form solution for the standard C¹ continuous cubic spline. It would seem appropriate to revert to that or similar. —Preceding unsigned comment added by PondTapir (talk • contribs) 04:12, 7 August 2010 (UTC)

Confused Algorithm for computing clamped cubic splines

The algorithm appears to compute the natural cubic spline, which is the title of the preceding section (and this one does not describe an algorithm). A clamped cubic spline is not otherwise defined in the article. Also, the values computed for the z array are never described. This is especially confusing because the array z is used in the description following as the second derivative at the knots and the values of z computed are not. Other computed terms could use descriptions as well, such as alpha (triple the difference of the mean slope on adjacent intervals). Further, the mean slope in each interval could be used to simplify the description of the computation of alpha. The array mu is allocated to hold n+1 items, but only n are assigned (or used). JCRVMD (talk) 23:01, 7 January 2011 (UTC)

Improvement of this article!

Why does the article start with this image? ? Of all spline images in "WIKI Commons" this seems to be the strangest and the least suitable for the article!

It says:

"In mathematics, a spline is a special function defined piecewise by polynomials."

and

"In computer science subfields of computer-aided design and computer graphics, the term "spline" more frequently refers to a piecewise polynomial (parametric) curve."

There is therefore no difference in the use of this term between "mathematics" and "computer science"!!

The "Introduction" and even more the "Definition" are strange making rather simple matter sound complicated with a lot of mathematical terminology that really does not help!

A re-write:

++++++++++++++++++++++++++++++++++++++++++++

In mathematics, a spline of degree n is a function defined in a number of adjacent intervals by polynomials of degree n, i.e.

$S(t)=P_{0}(t){\mbox{ , }}t_{0}\leq t<t_{1},$

$S(t)=P_{1}(t){\mbox{ , }}t_{1}\leq t<t_{2},$

$\vdots$

$S(t)=P_{k-1}(t){\mbox{ , }}t_{k-1}\leq t\leq t_{k}.$

where the polynomials $P_{0}(t),P_{1}(t)\cdot P_{k-1}$

are such that $S(t)$ is a continuous and (n-1) times differentiable function in the full interval $t_{0}<t<t_{k},$. A simple example of a spline of degree 2 is

$S(t)=(t+1)^{2}-1{\mbox{ , }}-2\leq t<0$

$S(t)=1-(t-1)^{2}{\mbox{ , }}0\leq t\leq 2$

for which $S'(0)=2$.

The by far most important spline functions are the "cubic splines" of degree 3 as these are the splines used for spline interpolation for which the 4 parameters of a third order polynomial in general are needed to get a good and smooth interpolation. The term "spline" in fact originate from the elastic splines used by craftsmen to draw smooth bent curves passing through a number of given point, called the "knots", and these curves can to a first approximation can be considered to be cubic splines where the "knots" are the points where the different third degree polynomials meet. For this, see Spline interpolation.

+++++++++++++++++++++++++++++++++++++++++++

And this should all be illustrated with figures!! The strange figure included could for example be replaced by a graph of the function

$S(t)=(t+1)^{2}-1{\mbox{ , }}-1\leq t<0$

$S(t)=1-(t-1)^{2}{\mbox{ , }}0\leq t\leq 1$

And actually I do not think much more should be said here. The (simple) definition is provided and the interesting part is then found in Spline interpolation, i.e the computation of the interpolating cubic spline!

Stamcose (talk) 19:55, 27 January 2011 (UTC)

Here is a complete rewrite removing a lot of stuff that just is confusing a reader who wants the basic stuff clearly and simply explained! Avoiding rather exotic generalizations "reducing the smoothness requirements" at the knots that if at all should be put at the very end of the article under the headin Generalizations of the concept! Who opposes this re-write wanting more of the old stuff to be kept and with what justification?

Proposed rewrite:

Definition

A spline is a piecewise-polynomial real function.

$S:[a,b]\to \mathbb {R}$

on an interval [a,b] composed of k ordered disjoint subintervals $[t_{i-1},t_{i}]$ with

$a=t_{0}<t_{1}<\cdots <t_{k-1}<t_{k}=b$.

On each interval i function S is a polynomial

$P_{i}:[t_{i-1},t_{i}]\to \mathbb {R}$,

so that

$S(t)=P_{1}(t){\mbox{ , }}t_{0}\leq t<t_{1},$

$S(t)=P_{2}(t){\mbox{ , }}t_{1}\leq t<t_{2},$

$\vdots$

$S(t)=P_{k}(t){\mbox{ , }}t_{k-1}\leq t\leq t_{k}.$

The order of the spline, n, is equal to the highest order of the polynomials $P_{i}(t)\quad i=1\cdots k$ used and the polynomials should be such that $S(t)$ is continuous and n-1 times derivable also at the inner knots points $t_{i}\quad i=1,\cdots k-1$ . This means that at all the inner knots points $t_{1},t_{2},\cdots ,t_{k-1}$ $P_{i}^{(j)}(t_{i})=P_{i+1}^{(j)}(t_{i})$ for all $j\quad 0\leq j\leq n-1$

If all subintervals are of the same length the spline is uniform if not it is non-uniform.

The by far most used splines are the cubic splines, i.e. splines of order 3, as these are used for spline interpolation simulating the function of flat splines

Examples

A simple example of a quadratic spline (a spline of degree 2) is

$S(t)=(t+1)^{2}-1{\mbox{ , }}-2\leq t<0$

$S(t)=1-(t-1)^{2}{\mbox{ , }}0\leq t\leq 2$

for which $S'(0)=2$.

An simple example of a cubic spline is

$S(t)=|t|^{3}$

as

$S(t)=t^{3}{\mbox{ , }}t\geq 0$

$S(t)=-t^{3}{\mbox{ , }}t<0$

and

$S(0)^{(0)}=S(0)=0$

$S(0)^{(1)}=S(0)'=0$

$S(0)^{(2)}=S(0)''=0$

For the drafting tool, see Flat spline.

See also: Spline interpolation

2. Example

<quote>[...] $S'(0)=2$ [...] $S(0)'=\ 0$</quote>
What's "$S(0)'$"? Is it supposed to be "$S'(0)$" similiar to the first example and accourding to the normal notation of the 1. derivate of S?
-91.63.238.36 (talk) 13:06, 12 December 2012 (UTC)

References

I added the external link to my free Web e-book. It is a more complete treatment of piecewise interpolation for those new to the subject and aimed at graphics. I'll add references to other related Wiki pages in the future.

EDIT Dec 29 2022 Comcast closed down web hosting. My Interpolation e-book appears to be available on the archive:

https://web.archive.org/web/20150930224506/http://home.comcast.net/~k9dci/site/?/page/Piecewise_Polynomial_Interpolation/ — Preceding unsigned comment added by EngineerSteve (talk • contribs) 00:27, 30 December 2022 (UTC)

My e-book is both a study of the fundamentals described in simple terms as well as a reference showing many types of Polynomial Interpolation -both common types and some developed by the author. It also shows some techniques not seen elsewhere. Linear interpolation is looked at carefully and shown is as the basis for all more advanced types using only Algebra. Only after understanding how adding squared and cubed terms cause smooth curves, are the more advanced curves examined such as Bezier, Catmul-Rom, b-spline, and Hermite. More advanced mathematical concepts and notations are kept to a minimum. Some additional techniques that are suggested by the mathematics and that the author has not seen elsewhere are examined. A reference is also included with all the most common curve drawing methods and includes drawings to allow comparison with other types.

WHY: I had hoped, but failed to find a book explaining polynomial interpolation basics and thought that one must certainly exist with a collection of interpolation types. I found either purely mathematical tests or advanced graphics texts. Several years later, I started reading the original Internet Usenet "groups", comp.graphics.algorithms, and did lots of searching on the net. I saved whatever I found related to splines and curves, but did not look at it or try to understand any of it until early in 1996, I decided to look at what I had collected, and started to figure things out. I begin recording my thoughts for future reference and this is the result. The very book I wanted originally is now freely available for others.

I do not believe this has any issues with the Wiki self cite guidelines. -- Steve -- (talk) 22:57, 28 August 2011 (UTC)

End re-write

Stamcose (talk) 13:09, 29 January 2011 (UTC)

It probably doesn't but inserting it as a link is worse since it's just self promotion without any attempt to improve this inline-reference-free mess of an article with anything, which is probably why somebody already deleted it. --A Shortfall Of Gravitas (talk) 02:14, 13 February 2025 (UTC)

Removal of dubious text

To my understanding this article is in a bad state! I already drafted above in "Improvement of article" what I think should be said. What is more delicate is what not should be said.

I think the text starting "Recall that a function.." should be removed! Seems to be completely misplaced and irrelevant. It says "is commonly denoted". Please give references showing that anything of this is "common" at all in connection with splines (in serious work!)!

The "Algorithm for computing natural splines" and "clamped splines" is also misplaced(nonsense!?) In this article the definition what a spline is should be given but there is nothing to compute! For "Spline interpolation" there is something to compute but this is done completely different!

I would like to remove all this dubious material! Any protests?

Stamcose (talk) 19:40, 29 January 2011 (UTC)

degree vs order

The degree of the polynomial p(x) = a_0 + a_1x + a_2x^2 + ... is the largest k for which a_k is not zero. In particular, the set of all polynomials of a given degree is not a linear vector space. For that reason, some people use the term "order" to indicate the number of consecutive terms, starting with a_0, a_1x, ... they are going to use to describe a polynomial. Thus, the set of all polynomials of order 4 is what some, carelessly, describe as the set of all cubic polynomials. The same people would call a spline of order 4 what, in this article, is called a cubic spline or a spline of order 3. It would be good if the article were not to confuse order with degree. Deboor (talk) 03:26, 26 June 2011 (UTC)

my spline algorithm creates a oscillating curve -- why?

I was of the opinion, that I understand spline interpolation. Last time I tried to implement this myself, I was successful in getting 0, 1, and 2nd derivative continuous. But the resulting spline was widely oscillating -- going through the given points but oscillating around them (0..2 derivative continuous)! Then I tried things like modifying existing free parameters (1th derivative at point 0) to minimize the integral of the square of the second derivative. Or even adding more coefficients to minimize this integral. But without success. Oscillation was reduced strongly but got again out of control in the last couple of intervals. Then I copied the implementation from Numerical recipes and everything was fine. I've to state, that I don't like the explanation here in wikipedia or the one in numerical recipes. Why not simply stating which things are given and what needs to be solved for?

given:

• x1..xn, y1..yn, n >= 3
• dy/dx continuous
• d^2y/dx^2 continuous
• polynomial for every interval is
• an+bn*x+cn*x^2+dn*x^3

or

• yn + bn*(x-xn) + cn*(x-xn)^2 + dn*(x-xn)^3

This makes n points given and n-2 midpoints where the 0, 1 and 2nd derivative must be identical. This is n+3*(n-2)=4n-6

need to be solved for:

• (n-1) intervals with 4 (polynomial) coefficients each, which is 4n-4 coefficients.

So we need two more given inputs -- maybe the 1 and 2nd derivative at the first point.

This gives a linear system with 4n-4 equations and 4n-4 unknowns.

Again: could somebody explain to me, what I'm doing wrong? Why is my solution oscillating? I don't see, how there could be more than one solution to this linear system.

18:35, 6 October 2015 (UTC)Elenchusis (talk) 18:35, 6 October 2015 (UTC) Hi there, I'm new to editing Wikipedia, so I'm not sure if this is where to respond, but... I believe the most common way of getting the two extra inputs you need is to set the 2nd derivative at the start and end points to zero. There are other methods, but I used this and was able to solve it just fine.

Misplaced text

User 71.246.201.83 on 13 April 2011 introduced some text (section "derivation") that in fact is the spline interpolation problem. But this is already available and much better formulated in the article Spline interpolation. I think this text should be removed, possibly replaced with some clear reference to the article Spline interpolation!

Opinions?

Stamcose (talk) 16:53, 18 September 2011 (UTC)

Yes, that makes sense: it doesn't really belong here, even without it already appearing in a far more appropriate article. Just replace it with a short paragraph and a {{main}} template.--JohnBlackburne^words_deeds 18:47, 18 September 2011 (UTC)

Preferred to as?

Suspect typo in second sentence - "preferred to as". I believe this should be "preferred to" due to the comparison to polynomial interpolation, but since the change could change meaning I am requesting confirmation from others before editing. This could also use a reference i.e. preferred by whom? — Preceding unsigned comment added by Jjensen2009 (talk • contribs) 00:37, 16 March 2012 (UTC)

'Smoothness' in introduction

The introduction to the article currently reads:

In mathematics, a spline is a sufficiently smooth polynomial function that is piecewise-defined, and possesses a high degree of smoothness at the places where the polynomial pieces connect (which are known as knots).

There is nothing about the general spline that needs to be 'smooth', however, for example a linear spline is in no way smooth at knots. I propose something to the extend of: "In mathematics, a spline is a piecewise-defined polynomial function which defines a curve or line connecting a series of points (which are known as 'knots'). A linear, or first-order, spline is one in which each point is connected to the next with a straight line. Higher order splines (e.g. quadratic, or 'second-order', splines) can be used to 'smooth' the connections between points." All Clues Key (talk) 05:10, 1 October 2012 (UTC)

What's the difference? (again)

With the advent of computers, splines first replaced polynomials in interpolation, ...

If splines are piecewise polynomials, this could be written better. —Tamfang (talk) 05:20, 27 September 2014 (UTC)

Spline page array index, readability of α and a

I copied this from my TP. Purgy (talk) 09:36, 5 March 2019 (UTC)

Hey, I notice you undid my edit, commenting that there is no $\alpha _{i-1}$ for $i=1$; however, all of the other arrays in this description appear to be indexed from zero, and in fact, $\alpha$ is actually zero-indexed in step one. Additionally, I have just implemented this algorithm in Common Lisp (my code can be found here), and it fails to work with $\alpha _{i}$ (as there is no $\alpha _{k-1}$), but works perfectly with $\alpha _{i-1}$. Goose121 (talk) 22:26, 4 March 2019 (UTC)

It's about this edit. It is correct that all of the other arrays in this description appear to be indexed from zero, (emphasis mine), but it is not true that $\alpha$ is actually zero-indexed in step one. Step one is about an array called a, which unluckily looks very similar to α. The array a is indexed from zero, but the array α is not. I will edit α to β to make the difference more obvious. I do not doubt that your implementation works, but I am convinced that the current description is correct, and maybe describes a different implementation of an equivalent algorithm. Purgy (talk) 09:36, 5 March 2019 (UTC)

Oops! I do admit that I confused $a$ with $\alpha$ in my original comment, and I guess that at some point I did end up subtracting 1 from the index when it was accessed in my implementation. So as it stands, the algorithm is still correct, however the mixing of zero- and one-indexed arrays is definitely confusing (and the one-indexing of $\beta$ doesn't even provide a significant simplification of the algorithm, seeing as $\beta$ is only accessed once, and making it zero-indexed would only require replacing $\beta _{i}$ with $\beta _{i-1}$). Goose121 (talk) 19:03, 5 March 2019 (UTC)

There is an encompassing index range [0, k]. With respect to the boundaries ranges of [0, k-1] (excluding the right rim) and [1, k-1] (excluding both rims) are necessary. Reindexing the latter to [0, k-2] might be appealing to index jugglers ;) , but, honestly, for the sake of tradition-bound zero-based indexing, I would not sacrifice the natural index range. In case there is really such an implementation of a language that reacts positively to such index manipulations, then, of course, it would be meaningful there to reindex, but not in this brute-force pseudo code, which should strive for better accessibility still.

Supplement for the semantics of the indices: currently the $k+1$ integers $j\in [0,k]$ enumerate the $k-1$ partioning points plus the two boundary points; the $k$ integer values $i\in [0,k-1]$ enumerate the generated intervals, the polynomials defined there, ...; and finally, the now not separately named integers $\in [1,k-1]$ refer to the $k-1$ inner (partitioning) points of the interval. Starting with [1, k+1] would generate [1, k] and [2, k], not necessarily a better variant. Currently I consider to start all ranges at one, giving the upper boundaries a meaningful interpretation: [1, k+1], [1, k], and [1, k-1], possibly frustrating traditionalist base-zero junkies, like E.W.Dijkstra. ;)

I just started to look over this article a bit, unifying the indices (i, j, n, perhaps, with fixed associated ranges) might be a step to improvement? Purgy (talk) 20:27, 5 March 2019 (UTC)

I renamed the degree of a spline from n to m, the number of intervals (polynomials) from k to n, and used the free k as index name for the range [1, n - 1]. The other ranges are now: j in [0, n] and i in [0, n - 1]. One could note that the i-range is used both increasing and decreasing under the same index name.

I intend to work on some places in the article and will collect impressions about the starting values of the ranges. Purgy (talk) 15:24, 7 March 2019 (UTC)

Do the piecewise curves actually equal $y_{i}$ ?

I'm reading the article to try to understand how splines are defined, and I ran across this equation in the section "Algorithm for computing natural cubic splines":

• $P_{i}\left(x_{i}\right)=y_{i}=P_{i-1}\left(x_{i}\right),i=1,\ldots ,n-1.$

But doesn't this mean that the generated function - or more precisely, the endpoints of every single piece of the function - fall directly on the input points? Sorry if I'm misunderstanding, I'm just learning about splines. Russl5445 (talk) 20:51, 17 March 2021 (UTC)

The pseudo code example is not helpful

I'm sorry but I would not call this pseudo code, a python or C example would be better. There are more explicit ways to write pseudo code. — Preceding unsigned comment added by Jokooon (talk • contribs) 22:05, 3 December 2021 (UTC)

Its been removed for WP:OR for now, unless somebody has a reference for it that isn't the author above posting it without knowing if it even works. Some better snippet of C or C++ can probably be found at one of the external links and converted to pseudocode. The fortran code link I left up while deleting the excess of them has Fortran90 files so big github refuses to show them in reader mode and they have to be viewed in raw, something like 6-9MB each which is insane. Computers back when that particular library was written could easily handle compiling multi-file projects bigger than that, so I don't know what was going on there. --A Shortfall Of Gravitas (talk) 02:14, 13 February 2025 (UTC)